DualCD {DebiasInfer}

Coordinate descent algorithm for solving the dual form of our debiasing program.

Description

This function implements the coordinate descent algorithm for the debiasing dual program. More details can be found in Appendix A of our paper.

Usage

DualCD(
  X,
  x,
  Pi = NULL,
  gamma_n = 0.05,
  ll_init = NULL,
  eps = 1e-09,
  max_iter = 5000
)

Arguments

X

The input design n*d matrix.

x

The current query point, which is a 1*d array.

Pi

An n*n diagonal matrix with (estimated) propensity scores as its diagonal entries.

gamma_n

The regularization parameter "\gamma/n". (Default: gamma_n=0.05.)

ll_init

The initial value of the dual solution vector. (Default: ll_init=NULL. Then, the vector with all-one entries is used.)

eps

The tolerance value for convergence. (Default: eps=1e-9.)

max_iter

The maximum number of coordinate descent iterations. (Default: max_iter=5000.)

Value

The solution vector to our dual debiasing program.

Author(s)

Yikun Zhang, [e-mail]

References

Zhang, Y., Giessing, A. and Chen, Y.-C. (2023) Efficient Inference on High-Dimensional Linear Model with Missing Outcomes. https://arxiv.org/abs/2309.06429.

Examples

  require(MASS)
  require(glmnet)
  d = 1000
  n = 900

  Sigma = array(0, dim = c(d,d)) + diag(d)
  rho = 0.1
  for(i in 1:(d-1)){
    for(j in (i+1):d){
      if ((j < i+6) | (j > i+d-6)){
        Sigma[i,j] = rho
        Sigma[j,i] = rho
      }
    }
  }
  sig = 1

  ## Current query point
  x_cur = rep(0, d)
  x_cur[c(1, 2, 3, 7, 8)] = c(1, 1/2, 1/4, 1/2, 1/8)
  x_cur = array(x_cur, dim = c(1,d))

  ## True regression coefficient
  s_beta = 5
  beta_0 = rep(0, d)
  beta_0[1:s_beta] = sqrt(5)

  ## Generate the design matrix and outcomes
  X_sim = mvrnorm(n, mu = rep(0, d), Sigma)
  eps_err_sim = sig * rnorm(n)
  Y_sim = drop(X_sim %*% beta_0) + eps_err_sim

  obs_prob = 1 / (1 + exp(-1 + X_sim[, 7] - X_sim[, 8]))
  R_sim = rep(1, n)
  R_sim[runif(n) >= obs_prob] = 0

  ## Estimate the propensity scores via the Lasso-type generalized linear model
  zeta = 5*sqrt(log(d)/n)/n
  lr1 = glmnet(X_sim, R_sim, family = "binomial", alpha = 1, lambda = zeta,
               standardize = TRUE, thresh=1e-6)
  prop_score = drop(predict(lr1, newx = X_sim, type = "response"))

  ## Solve the debiasing dual program
  ll_cur = DualCD(X_sim, x_cur, Pi = diag(prop_score), gamma_n = 0.1, ll_init = NULL,
                  eps=1e-9, max_iter = 5000)